Embedded newform 116.2.l.b.55.11

• Label: 116.2.l.b.55.11
• Level: $116$
• Weight: $2$
• Character: 116.55
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.l (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			55.11
Character	\(\chi\)	\(=\)	116.55
Dual form			116.2.l.b.19.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35022 - 0.420592i) q^{2} +(1.30343 - 0.456089i) q^{3} +(1.64620 - 1.13579i) q^{4} +(-3.64127 + 0.831096i) q^{5} +(1.56809 - 1.16403i) q^{6} +(-0.00558440 + 0.0115961i) q^{7} +(1.74504 - 2.22595i) q^{8} +(-0.854590 + 0.681513i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{28}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35022	−	0.420592i	0.954752	−	0.297404i
							\(3\)	1.30343	−	0.456089i	0.752534	−	0.263323i	0.0733651	−	0.997305i	\(-0.476626\pi\)
0.679169	+	0.733982i	\(0.262340\pi\)
\(5\)	−3.64127	+	0.831096i	−1.62843	+	0.371678i	−0.936604	−	0.350391i	\(-0.886049\pi\)
							−0.691822	+	0.722068i	\(0.743192\pi\)
\(7\)	−0.00558440	+	0.0115961i	−0.00211070	+	0.00438292i	−0.902022	−	0.431691i	\(-0.857917\pi\)
							0.899911	+	0.436074i	\(0.143631\pi\)
\(11\)	0.482244	+	4.28004i	0.145402	+	1.29048i	0.828432	+	0.560090i	\(0.189233\pi\)
							−0.683030	+	0.730390i	\(0.739338\pi\)
\(13\)	−1.91361	−	1.52605i	−0.530739	−	0.423250i	0.321107	−	0.947043i	\(-0.395945\pi\)
							−0.851846	+	0.523793i	\(0.824517\pi\)
\(17\)	0.542577	+	0.542577i	0.131594	+	0.131594i	0.769836	−	0.638242i	\(-0.220338\pi\)
							−0.638242	+	0.769836i	\(0.720338\pi\)
\(19\)	1.93526	−	5.53065i	0.443978	−	1.26882i	−0.476897	−	0.878959i	\(-0.658238\pi\)
							0.920875	−	0.389858i	\(-0.127476\pi\)
\(23\)	5.63038	+	1.28510i	1.17402	+	0.267961i	0.764672	−	0.644419i	\(-0.222901\pi\)
							0.409343	+	0.912381i	\(0.365758\pi\)
\(29\)	−5.34431	−	0.662064i	−0.992414	−	0.122942i
							\(31\)	−6.05567	−	3.80503i	−1.08763	−	0.683404i	−0.136019	−	0.990706i	\(-0.543431\pi\)
−0.951612	+	0.307302i	\(0.900574\pi\)
\(37\)	−3.39523	−	0.382551i	−0.558173	−	0.0628909i	−0.171628	−	0.985162i	\(-0.554903\pi\)
							−0.386544	+	0.922271i	\(0.626331\pi\)
\(41\)	0.383115	−	0.383115i	0.0598325	−	0.0598325i	−0.676557	−	0.736390i	\(-0.736529\pi\)
							0.736390	+	0.676557i	\(0.236529\pi\)
\(43\)	0.0943772	+	0.150201i	0.0143924	+	0.0229054i	0.853843	−	0.520531i	\(-0.174266\pi\)
							−0.839451	+	0.543436i	\(0.817123\pi\)
\(47\)	8.20811	−	0.924832i	1.19728	−	0.134901i	0.509271	−	0.860606i	\(-0.329915\pi\)
							0.688005	+	0.725705i	\(0.258486\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.55.11	yes	144
4.3	odd	2	inner	116.2.l.b.55.5	yes	144
29.19	odd	28	inner	116.2.l.b.19.5	✓	144
116.19	even	28	inner	116.2.l.b.19.11	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.19.5	✓	144	29.19	odd	28	inner
116.2.l.b.19.11	yes	144	116.19	even	28	inner
116.2.l.b.55.5	yes	144	4.3	odd	2	inner
116.2.l.b.55.11	yes	144	1.1	even	1	trivial